\section{Tangent Vectors}
\begin{defn}
\marginpar{Algebraic Sets often taught as ``surfaces'' in calculus courses in universities}Let $S\subseteq\mathbb{R}^n$ be a set. If our set $S$ is the locus of zeros of one or more polynomial functions $f(x_1,\ldots,x_n)$, it is called a \textbf{real algebraic set}:
\begin{equation}
S = \{ x\in\mathbb{R}^n: f(x) = 0\}.
\end{equation}
\end{defn}

\begin{ex}
The set $S\subset\mathbb{R}^3$ defined by the zeros of the polynomial
\begin{equation}
f(x,y,z) = x^2 + y^2 + z^2 - 1
\end{equation}
defines a sphere of radius 1. QEF.
\end{ex}

\begin{ex}
The set $S\subset\mathbb{R}^3$ defined by the zeros of the polynomial
\begin{equation}
f(x,y,z) = \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1
\end{equation}
(where $a,b,c\in\mathbb{R}^{+}$) defines an ellipsoid. QEF.
\end{ex}

\begin{lem}\label{lemma1}
Let $S$ be a real algebraic set in $\mathbb{R}^n$, defined as the locus of zeros of one or more polynomial functions $f(x)$. The tangent vectors to $S$ at $x$ are orthogonal to the gradients $f(x)$.
\end{lem}
